Introduction

The demand for mathematics graduates in Australia is ever escalating due to the central role of mathematics in science, technology, and engineering (McLaughlin et al., 2015; Reid et al., 2016; Sharma & Yarlagadda, 2018). Unfortunately, in the last two decades, Australia has experienced a decrease in both mathematical performance (Organisation for Economic Co-operation and Development [OECD], 2019), and the number of students studying advanced mathematics at school and university (Hine, 2019). Despite diminishing numbers of high school students studying advanced mathematics, there is limited data explaining this decline (Kennedy et al., 2014). One consideration to explain such trends is that success and understanding of key concepts in mathematics in junior secondary years (Years 7 to 10) is likely to be an important consideration for students in selecting pathways that lead to the study of advanced mathematics in senior years of secondary schooling (Years 11 and 12) (Bong, 2013).

In Australia, the study of quadratic equations has been positioned towards the end of Year 10 in curriculum (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2018), which is when students are making decisions regarding the study of more abstract senior mathematics subjects (e.g., Mathematical Methods, and Specialist Mathematics) designed in preparation for STEM-orientated tertiary studies, or the less abstract strands (e.g., Essential Mathematics, and General Mathematics). Following Year 10, a review of quadratic equations is included in the Year 11 Mathematical Methods curriculum (Queensland Curriculum and Assessment Authority [QCAA], 2019). The review of quadratics is one of five topic areas to be undertaken within approximately 20 h of the scheduled teaching in the Mathematics Methods course. As noted above, success with quadratic equations is likely to act as a gatekeeper to studying further mathematics in senior secondary school, since students who are confounded by the algebra associated with quadratics in Year 10 are likely to experience challenges when faced with the more abstract algebra that underpins calculus (Edge & Friedberg, 1984; Moore, 2005; Pyzdrowski et al., 2013). Given that it has been acknowledged that “the most important factor in determining success in calculus was manipulative skills in algebra” (Edge & Friedberg, 1984, p. 137), difficulties with quadratics in junior secondary years may predispose students to not commence the study of further abstract algebra that is included in the Mathematical Methods and Mathematics Specialist senior mathematics subjects in Australia. This is confirmed in the Australian curriculum documents as the rationale for Mathematical Methods subject notes that “the major themes of Mathematical Methods are calculus and …include as necessary prerequisites studies of algebra” (ACARA, 2018). Further, Specialist Mathematics notes that knowledge and skills from the Year 10 content descriptor “investigate the concept of a polynomial, and apply the factor and remainder theorems to solve problems” (ACARA, 2018) are highly recommended in preparation for the course. This prerequisite recommendation was one of only three made for the course (with the other two focusing on trigonometric concepts).

Bong (2013, p. 64) explains the importance of success with prerequisite concepts concisely: “Direct mastery experience is the most reliable source (of self-efficacy). Success with tasks raises self-efficacy towards it, whereas failure lowers it”. Self-efficacy has been strongly correlated with outcome attainment including subject selection (Burton, 2004). Therefore, one purpose of this study was to identify knowledge barriers that students may have that prevent or discourage them from studying more advanced senior mathematics. The analysis of students’ struggles with quadratics offered insight into the knowledge forms that were potentially acting as cognitive, and subsequently attitudinal barriers to further study of advanced mathematics.

The nature and implementation of the curriculum are a central aspect of student success or otherwise, not least because curriculum sets the academic goals and timeframes for their achievement. Stein et al. (2007) noted that the nature and implementation of mathematics curricula was a critical aspect of mathematics teaching and learning and warranted investigation. Similarly, Brosnan et al., (2013, p. 349) reported: “Decades of research and though have been directed toward developing successful curriculum for school mathematics”. Thus, this study continues that process by examining student success and, by using historical school-based contextual data and published curriculum recommendations, seeks to understand some of the curriculum implementation factors that potentially impact on student outcomes. If students have failed to master the necessary content, is it possible that the curricula structure is a contributing factor? Stein et al. (2007) noted that time allocation was an important variable and “flexibility in class scheduling and timing (was) a luxury infrequently afforded teachers.” (p. 355).

The primary aim of the paper is to explore student’s success with quadratic equations, and subsequently identify any persistent barriers to success in more advanced mathematics. A secondary aim is to reflect on the structure of the curriculum and the way it was implemented as a potential explanation for some of the difficulties the students experienced.

Literature review

Overview of research relating to solving quadratic equations

One challenge in the middle years of secondary schooling is to meaningfully develop secondary students’ conceptual understanding of non-linear functions including quadratics (Fonger et al., 2020). The resounding theme in research is that students’ performance in the domain of quadratic equations is poor and does not significantly increase after detailed instruction (Chaysuwan, 1996; Vaiyavutjamai & Clements, 2006; Vaiyavutjamai et al., 2005). Authors have proposed numerous reasons for this including an overemphasis on some solving techniques such as factorisation due to poor fractional and radical arithmetic skills (Bosse & Nandakumar, 2005), key misconceptions concerning the concepts of variables and the meaning of ‘find a solution’ (Vaiyavutjamai & Clements, 2006; Vaiyavutjamai et al., 2005), and overemphasis on either procedural or conceptual understanding when teaching the topic (Kotsopoulos, 2007; Vaiyavutjamai & Clements, 2006). The challenges with the study of quadratics have also been found to extend to teachers in the middle secondary years (Huang & Kulm, 2012). However, beyond the confirmation of student difficulties, there is a general deficit in empirical evidence explaining students’ difficulties in this specific topic, especially in the Australian context.

Mathematics curriculum structure

The current Australian Curriculum for mathematics is outlined and enacted as a spiral curriculum that is hierarchical in nature. In Fig. 1, it can be observed that the topic of quadratics is progressively built up to from Year 5 onwards, and similar and related topics are revisited at increasing depth in subsequent years.

Fig. 1
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The hierarchical development of skills and knowledge related to quadratic equations in the Australian curriculum

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A critical assumption of hierarchical mathematics curricula is that the early stages lay a conceptual understanding upon which to build increasing complexity and abstraction. The suggested benefits of this have included the enabling of integration and connection between concepts, moving away from a compartmentalised view of mathematical concepts (Harden, 1999). The organisation of a hierarchical mathematics curriculum aligns with what is understood with schema development, which emphasises the importance of linking new knowledge to existing schema (Van Merrinboer & Pass, 1990). The intention of organising mathematics curriculum in such a manner is that topics are revisited, and new learning is related to previous learning which ideally contributes to increasingly complex schema development (Harden, 1999). Piaget (1960) and Skemp (1976) supported the idea that relationships between ideas and procedures, schemas, are reformed when there is an element of cognitive dissonance. The struggle to make sense of the new information prompts the re-organisation of schema into more powerful models. Underpinning this assimilation of knowledge is the assumption that the learner has the cognitive tools and necessary scaffolding that make sense of the new information.

In terms of schema development and the way in which knowledge builds upon knowledge, the critical role of prior success in predicting future success has been well documented by Hattie (2008) who cites an effect size of 0.94 for prior achievement. Unfortunately, it is well known that people forget, meaning that the neural pathways that constitute memories decay or are displaced (Della Salla, 2010). Memory loss can be averted if the thought is embedded in schema and if used repeatedly (Sweller, 2016; Van Merrienboer & Pass, 1990). Hattie and Zierer (2019, p. 82) defined deliberate practice as conscious practice that was challenging, varied and regular and had a positive effect on learning (d = 0.49) due to the strengthening of long-term memory. The importance of repetition that develops automaticity is recognised in the Australian Curriculum as the proficiency strand fluency (ACARA, 2018). Similarly, schema or the connection of knowledge into general categories is recognised in ACARA (2018) description of understanding in the proficiency strands.

For students in mathematics, sequences of learning where the level of difficulty increases at each successive revisiting of a topic relates strongly to the iterative relationship between conceptual understanding and procedural fluency (Rosenshine, 2012; Sfard & Linchevski, 1994). Skemp (1976, p.9) use the term relational understanding, and defined it as “knowing both what to do and why”. He contrasted this with instrumental understanding, “rules without reason”. Skemp noted that relational understanding took more time and scaffolding to develop. Hiebert and Lefevre (1986) used the term conceptual understanding, which is characterised most clearly as deep understanding of the relationships between critical pieces of information, the equivalent of schema. The intention in mathematics is to teach conceptually, followed by establishing procedural fluency (Kamii & Dominick, 1998; National Council of Teachers of Mathematics [NCTM], 2014). The next time the topic is then revisited, the intention is then for a deeper conceptual understanding to be developed. However, if the new concept is not well integrated into schema and if there is limited repetition to aid remembering, the knowledge is likely to be held tenuously in long-term memory and forgotten.

An additional potential issue associated with the organisation of mathematics curricula relates to the assumption that teaching of concepts in the following year or years begins from where students left off. It assumes that students had adequate experience, time, and exposure to the concept in prior years to develop fluency and mastery, and did not forget in the interim. It also assumes and relies on students being continuously part of the same academic program (e.g., remaining at the same school or not omitting key stages in prior years) (Gibbs, 2014). With a reportedly overcrowded mathematics curriculum in Australia (Donnelly & Wiltshire, 2014), there is a potential problem that teachers cover many topics briefly and without depth (Snider, 2004). This can result in a lack of sufficient conceptualisation and procedural fluency, which acts to limit understanding of the subsequent learning and exacerbates the possibility of forgetting.

Mastery approaches

A mastery learning approach to structuring and organising curriculum has been shown to have a significant influence on student achievement. Mastery learning, an approach identified as effective in meta-analyses, was found to have a positive influence on student attitudes as well as achievement (Hattie, 2008; Kulik et al., 1990), which is a highly sought-after educational outcome often difficult to obtain.

Mastery learning initially arose from models such as Bloom’s Learning for Mastery (Bloom, 1968). Bloom’s model focused on individualising instruction depending on student’s needs resulting in uniformly high performance for all. This model differentiated from the conventional models typically seen in some explicit instruction approaches where students receive identical instruction irrespective of individual needs, resulting in normally distributed outcomes within a class (Kulik et al., 1990). Mastery learning is structured around the design of small units of sequenced work developed from pre-testing on unit objects, followed by instruction on objects which are yet to be achieved (Hattie, 2008; Willett et al., 1983). Timeframes do not constrain the units of work, and the premise of mastery learning is that no student will progress onto further units until mastery of objectives is obtained. Hattie (2008) rated the effect size for mastery learning at d = 0.58 which is in the zone of desired effects. From a theoretical perspective, mastery learning is supported by Piagetian thinking that subsequent learning be built upon the foundation of earlier understandings.

A critical component of a mastery approach is the pretesting prior to a unit of work. In mathematics, this is frequently referred to as diagnostic testing. Diagnostic testing has long been recognised as an important component of teaching in mathematics, forming a cyclical approach to teaching and learning (Ashlock, 1976). Diagnostic testing facilitates the identification of students’ capabilities (current achievement) and allows the teacher to subsequently hypothesise potential reasons for students’ difficulties, formulate objectives to structure remediation of difficulties, and employ corrective remedial procedures in a cycle of ongoing evaluation (Reid O’Connor, 2020; Glennon, 1963; Mager & Peatt, 1962; Popham & Baker, 1970; Reisman, 1977, 1982). Therefore, diagnostic testing is one way in which error analysis is implemented in the design of units of work to facilitate a mastery approach in mathematics. Such testing also supports a Piagetian approach to cognitive acceleration where pre-testing allows the planning of appropriate challenge and support (Adey & Shayer, 2013).

Whether the current design of the Australian curriculum facilitates students obtaining mastery of mathematical concepts and skills, or at least sufficient prerequisite knowledge to attempt the next stages of learning, is an important educational consideration. This study is significant for the insight it provides into the impact of curriculum design in relation to student achievement for key mathematics concepts such as quadratic equations.

The study of quadratics in the Australian curriculum

Structuring curricula in a hierarchical manner intends for students to build understanding by extending the complexity and depth of which the topic is explored each time the concept is reviewed. This can be observed in the advice for the study of quadratic equations in the Australian Curriculum, Assessment and Reporting Authority (2018). As outlined in Fig. 1, students begin the study of formal algebra in Year 7 (i.e., the concept of variables is introduced). This is followed by exploration of the distributive law in algebraic contexts in Year 8, with the expectation that this builds off students’ understanding of the distributive law in whole number contexts from Year 7 (ACARA, 2018). Year 8 students are also introduced to factorisation of simple algebraic expressions by removing whole number factors, and they explore the links between factorisation and expansion. In Year 9, the distributive law is applied to the expansion of binomial products and parabolas are explored in graphical contexts. Year 10 students build on factorisation skills developed in Year 8 and explore the factorisation of expressions by removing common algebraic factors, and factorising monic quadratics. It is specifically noted in Year 10 that students also solve quadratics using a variety of strategies such as completing the square or the quadratic formula. In Queensland, the state specific curriculum implemented in the sample school at the time of this study (see Table 1) included the study of quadratics within a Year 10 unit on linear and non-linear relationships (Department of Education, Training and Employment [DETE], 2013).

Table 1 Suggested scope and sequence of the Year 10 linear and non-linear relationships unit as detailed in the Curriculum into the Classroom unit (DETE, 2013)
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Within this 12-lesson unit, quadratics were allocated to four out of 12 lessons (~ 4 h of teaching time). Of these four lessons, factorisation of monic quadratics was heavily emphasised (2 lessons), with all other methods of solving being delegated to the final two lessons of the unit.

The study of quadratics in international curricula

To give context to the sequencing of content related to quadratics in the Australian curriculum, it is useful to observe how international curricula address this topic. Commonly, factorisation is the first method taught when solving quadratics (e.g., Hong Kong curriculum: Hong Kong Special Administrative Region Government [HKSARG], 2007; Queensland curriculum: DETE, 2013; Singapore curriculum: Singapore Ministry of Education, 2012). However, earlier literature suggested that a cause of students’ limited success with quadratic study might be the overemphasis of some procedures such as factorisation (Bosse & Nandakumar, 2005). Such a critique is consistent with the notion that there has been too much emphasis on procedural knowledge at the expense of conceptual knowledge in mathematics (e.g., Rittle-Johnson et al., 2015). However, when comparing Australia’s curriculum to other top-performing nations (e.g., Hong Kong and Singapore as demonstrated by recent TIMSS testing; Mullis et al., 2016), there are important differences in how quadratic equations are approached. Like Australia, in the Hong Kong middle school curriculum, quadratics are first solved using factorisation. However, once students learn factorisation, they are then purposefully exposed to quadratics with irrational roots that are difficult to factorise. It is suggested that the intent of this is to immediately induce productive struggle and prompt a deep conceptualisation of the topic area.

There are also important differences in the amount of time allocated to learning quadratic equations in international curricula. In the Hong Kong curriculum, a total of 19 h is dedicated to learning how to solve quadratics through factorisation, graphing, and the quadratic formula. Further, in Singapore, learning quadratics occurs over an extended timeframe. Instead of learning quadratics in a single school year (as is typically done in Australia), Singaporean students learn how to solve quadratics through factorising in one school year, and in the next year they learn how to solve using the quadratic formula, completing the square, and graphing. The extended timeframe exhibited in Hong Kong curriculum, and the extended duration of focus on quadratics in Singapore contrasts with the compartmentalisation of quadratics to Year 10 and the short timeframes allocated to learning factorisation and other solving techniques in the Queensland specific curriculum as outlined above (DETE, 2013).

Aims of the study

This study aimed to describe student performance in solving quadratic equations, and identify patterns in the types of errors or misconceptions that were hindering students’ success. Meeting this aim allowed for key error patterns to be described for the sample. From this, the effect of the current mathematics curriculum, as enacted in this school, can be inferred and recommendations and conclusions for mathematics curriculum design can be made. The generalisability of the results to broader school settings can be inferred by the reader from the description of the sample.

Method

Overview of methodology

This study was part of a larger project collating data on students’ abilities to solve quadratic equations (see Reid O’Connor & Norton, 2016 for other findings). As the purpose of the study was to understand why students had difficulties solving quadratic equations, the methodological approach was exploratory (Creswell, 2012). The topic area of quadratics was chosen because it draws on an understanding of most algebra conventions studied in earlier grades. There were two phases of data collection; a written test and this was followed by diagnostic interviews to add nuance understandings for student’s written responses.

The first stage of the project involved collecting students’ written responses to a set of mathematical problems. These responses were analysed and the generation of categories explaining students’ errors was carried out from the perspective of an emergent design rather than beginning with predetermined categories. The literature review provided analytical guidelines in relation to prevalent difficulties, but these were not fixed and trends in the data were allowed to emerge. The data analysis was inductive, moving from specific analysis of individual student errors to a broad comparison, meaning that comparison happened from error pattern to error pattern, error pattern to categories of errors, and categories of errors to other categories.

The second stage of the project involved constructing interview questions that were designed in response to the results observed in the written test. The questions were aimed at ascertaining students’ reasoning behind some of the strategies employed and assessed whether students possessed a conceptual understanding of the nature of quadratics, solving techniques, and the meanings of the solutions. This data helped to triangulate the findings that emerged from the analysis of the written scripts. A critique of curriculum documents provides a background to be considered when interpreting the results.

The sample

The sample school in this research project was a coeducational high school in Queensland in a community of mixed socio-economic index. The school is typical of outer suburban schools according to MySchool data (ACARA, 2012). The sample comprised a Year 11 Mathematics B class. Mathematics B classes generally consist of students that have completed Year 10 mathematics to a high standard and, thus, have chosen to complete a mathematics subject that will contribute to their tertiary entrance and also gain them the prerequisites to particular STEM focused university courses. The course involves the study of further advanced algebra, and includes introductory calculus concepts. Mathematics B classes also consist of some students that have elected to study an additional higher level, extension mathematics subject (Mathematics C). The class consisted of 25 students, 11 of whom were Mathematics B only students, and 14 of whom were Mathematics B and C students. Therefore, over half of this class had elected to study the highest level of mathematics in the senior secondary years. The entire sample (n = 25) participated in the written test, and 14 students from this sample participated in the interviews. The selection of students who were interviewed depended on attendance on the day that the interviews were conducted; all available students were interviewed.

This sample was chosen because it was considered that it could give insight into the implementation of the state curriculum and the readiness of students to undertake advanced mathematics study in the final 2 years of school mathematics. All the students had studied quadratics in Year 10 consistent with the state and national curriculum (ACARA, 2018; DETE, 2013; Queensland Studies Authority [QSA], 2004) and were about to briefly revise the topic area before embarking on the study of differentiation, a critical introduction to calculus. Quadratics were taught in Year 10 consistent with the state curriculum at the time (outlined in Table 1). The sample school noted that solving quadratic equations using the quadratic formula was not emphasised in the units related to quadratic equations. In this study, the teaching of quadratics in Year 10 was not observed, however, the local curriculum as described in Table 1 provides insight into the types of learning experiences for this group of students.

Testing instruments

Students’ procedural and conceptual knowledge of key concepts associated with studying quadratic equations was probed using a pencil-and-paper test. Written tests have been utilised in previous studies concerning quadratic equations (e.gVaiyavutjamai & Clements, 2006; Vaiyavutjamai et al., 2005; Zakaria et al., 2010). The structure of the test was informed by previous literature on students’ challenges associated with studying quadratics, and local textbook and curriculum documents. Conditions of the test included that all solutions be written, and calculators were not permitted as it was important to determine whether fundamental mathematics, including basic number computations, was a factor limiting student achievement. Such a constraint is consistent with the curriculum as students must be fluent with and without the use of graphing calculators (ACARA, 2018; QSA, 2004). This paper examines the results of student responses to five of the test questions. The test questions are outlined in Table 2.

Table 2 Test questions
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The results of the written test were analysed qualitatively by noting any and all errors in students’ work. The errors were then categorised, and common themes found. The percentage success and fail rate and number of error occurrences were also recorded during the analysis.

Analysis and error classification

Below is a detailed description of the research team’s analysis of a student’s response to a test question. The research team (authors) classified and coded errors collaboratively. The original sample of work is shown in Fig. 2. The following breakdown in Fig. 3 provides insight into how the final error summary was obtained.

Fig. 2
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Example of unannotated student work

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Fig. 3
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Example of research team’s error analysis from student work

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In the sample of student work shown in Fig. 3, there is a key conceptual error related to the topic of quadratics (the first error identified), which was that the procedure carried out was unnecessary. The subsequent errors (the student’s inability to rearrange the equation correctly) are essentially procedural as they involve errors in basic prerequisite knowledge and understanding that ought to have been understood and committed to long-term memory in earlier years for fluent retrieval and use. In other words, the student lacked foundational algebraic fluency. It can be seen that some errors arising from the inability to apply given procedures (e.g., applying the null factor law) are classified as conceptual, as they are indicative of a deficit in understanding of the key concept, rather than a misapplication or error associated with a known procedure. The student in the above example has made no attempt to employ the null factor law, meaning that the correct procedure has not been identified. Thus, this has been classified as a lack of conceptual understanding relating to the nature of quadratic equations. Overall, the student whose work is shown in this example has not understood that the equation is a quadratic, has not applied an appropriate solving technique, and has misapplied algebraic procedures.

Analysis of this form has the potential to assist in developing a deep understanding of the reasons why students struggle with this area of mathematics study, and to inform future corrective measures. This method of analysis shows the transparency between the data and the error classification. This gives confidence in the inductive approach to classifying errors.

Results and analysis

Overview of data

The data is presented first as a description of success or lack of success on each question. The written scripts were then analysed for themes and finally, interview data is presented that adds nuanced meaning to student inscriptions.

In Table 3, the overall results for each question are outlined. The data indicates the challenges experienced by most students.

Table 3 Number and percentage of students who answered questions successfully, unsuccessfully, or did not attempt from a sample of 25
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Overall, the data illustrates that students’ abilities to solve quadratic equations were poor. It is evident that students’ difficulties increase as the questions become more complex, particularly when students are required to factorise where a ≠ 1. The high proportions of students who did not attempt Questions 3, 4, or 5 suggest that students did not know any method of solving the quadratic. This inference was triangulated with the interview data. Students simply made statements such as “I did not know where to start”. The student responses to the questions are explored in detail below.

Equations of the form \(\left({\varvec{x}}-{\varvec{r}}\right)\left({\varvec{x}}-{\varvec{s}}\right)=0\)

The solution to an equation of the form \(\left(x-r\right)\left(x-s\right)=0\) stems from the null factor law which states that, if the above equation is true, \(x-r=0\) or \(x-s=0\) and thus, \(x=r\) or \(x=s\). In solving the factorised quadratic, one type of error made by students was attempting to expand the equation to solve. This suggests that students did not identify the question as a quadratic or did not understand how to apply the null factor law in this scenario. Both errors essentially demonstrate a lack of conceptual understanding in regard to quadratics and their forms, as well as the null factor law as there is no logic in expanding a factorised quadratic to solve. Evidence for these errors is presented in Fig. 4.

Fig. 4
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Example of student demonstrating that they did not identify the question as a quadratic equation and did not use appropriate methods to solve

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Other errors made in this question, such as those exemplified in Fig. 5, further indicated that some students had no understanding or recognition of equations of this form, or the null factor law. Further analysis of the data also demonstrated that all students who were unable to answer this question were also unable to answer Questions 2, 3, and 4 involving solving quadratics in standard form.

Fig. 5
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Example of student averaging the values of 3 and 5 to obtain an answer of 4

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Interviews with students also concurred with the findings from the analysis of test scripts. Arising from the types of difficulties exhibited in the test, students were asked what type of equation \(\left(x-3\right)\left(x-5\right)=0\) was, what does the solution to the quadratic equation \(\left(x-3\right)\left(x-5\right)=0\) give you, and how do you know that if \(\left(x-3\right)\left(x-5\right)=0\) that \(x=5\) or \(x=4\) (checking for understanding of the null factor law). It was confirmed that approximately three quarters (10) of the interviewed students (n = 14) could not identify the given equation as a quadratic. The most common response was “I don’t know”. Students who responded in this brief manner resisted further verbal probing to explain their understanding. The interview data supports the findings from the written tests.

For the solution to the equation, all but one of the interviewed students could not identify that the solution gives the x-intercepts with answers ranging from “the point on the graph”, “the answer”, “it gives you how to solve”, to “not sure”. This confirms students’ lack of conceptual understanding regarding the nature of quadratics and the solution. Similarly, just over half of the interviewed students could not demonstrate conceptual understanding of the null factor law in interviews. Answers varied from “I only know to pick the two numbers” (indicating potentially a procedural understanding, but lacking a conceptual understanding of the law), to “something to do with other information given” (indicating a lack of procedural or conceptual understanding of the law).

Students’ difficulties in solving the equation \(\left(x-3\right)\left(x-5\right)=0\) were also confirmed in two larger scale studies by Vaiyavutjamai and Clements (2006) and Vaiyavutjamai et al. (2005). These authors found that even students who may have initially obtained correct answers to the question still lacked conceptual understanding of quadratics and essentially “did not know what they were talking about” (Vaiyavutjamai & Clements, 2006, p. 73). These authors found similar errors as exhibited by this sample of students, particularly with the large portion of students expanding the brackets first rather than solving the already factorised expression (Vaiyavutjamai et al., 2005).

Equations of the form \({{\varvec{x}}}^{2}+{\varvec{b}}{\varvec{x}}+{\varvec{c}}=0\) and \({{\varvec{x}}}^{2}+{\varvec{b}}{\varvec{x}}-{\varvec{c}}=0\)

There were two questions on the written test involving quadratics in standard form where \(a=1\) on the written test. Question 2 specifically requested that the student factorise the quadratic, but students were allowed to solve Question 3 using any method. It was a point of interest to observe what method the students preferred in Question 3 and later questions.

It was found that half of the sample was successful in solving Question 2. The written solutions of unsuccessful students indicated a variety of misconceptions. Four of the tested sample were able to factorise the equation but were not able to apply the null factor law to obtain a final answer to the question. It was found that the four students who gave these answers were also all unable to complete Question 1, which was finding solutions to a quadratic equation in factorised form. Relating to application of the null factor law, another error was only quoting one solution for the equations given in Question 2 and 3. This is potentially due to a lack of understanding regarding the null factor law. The example in Fig. 6 illustrates the student’s assumption that there can only be one solution, which is typical of linear equations.

Fig. 6
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Example of a student not realising that a quadratic equation has two solutions

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These findings concur with the types of errors found in Question 1 that were confirmed in the interview; students in the sample had difficulties identifying a factorised equation as a quadratic and did not possess a deep conceptual understanding of what the solutions meant.

Similar to Question 1, it was also observed that students attempted to rearrange the equation as they would when attempting to solve a linear equation. This suggested that students did not identify the question as a quadratic. In rearranging these equations, students demonstrated a large array of algebraic misconceptions as well. An example of this is outlined in Fig. 7.

Fig. 7
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Example of student incorrectly attempting to rearrange the equation

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In interviews, students were asked to identify the type of equation that \({\mathrm{x}}^{2}+9\mathrm{x}+20=0\) represented, and just over three quarters of the sample (11 out of 14) were unable to identify the equation as a quadratic. There three responses provided when asked what type of equation \({\mathrm{x}}^{2}+9\mathrm{x}+20=0\) was were “expand” (1 student), “not sure” (10 students), or “quadratic” (3 students).

In Question 3, a third of the sample was successful. For this question, students were able to select their preferred method of solving and it is a point of interest to note that none of the students attempted to use the quadratic formula, and all students who attempted the question used factorisation. However, 14 students from the sample (just over half) were unable to factorise the quadratic at all. Of the students who attempted to factorise, all were able to obtain the correct factorised form but only three quarters of these students were able to apply the null factor law correctly and obtain two correct solutions. This is an important finding as it demonstrates that most students who are procedurally fluent with factorisation still lack a deep understanding of the null factor law and how it is applied to solving binomial products, a finding that was confirmed in the interviews with students. It is reasonable to expect high levels of success with this question as \(a=1\), which meant that the required factorisation process was relatively simple. Despite this, misapplication and a lack of understanding of the null factor law appears to have impeded many students’ success across many different test questions. This demonstrates the critical nature of conceptual understanding regarding the meaning of this rule. Students do not appear to be aware that they are seeking the values of \(x\) where \(y\) is 0 (the x-intercepts).

Similar errors to those observed in Question 2 were observed in Question 3. As demonstrated in Fig. 8, students from the sample were unable to apply the null factor law in order to obtain solutions, despite being able to factorise the equation correctly.

Fig. 8
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Example of student finding the correct values for factors but incorrect signs

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Equations of the form \({{\varvec{a}}{\varvec{x}}}^{2}+{\varvec{b}}{\varvec{x}}+{\varvec{c}}=0\) and \({{\varvec{a}}{\varvec{x}}}^{2}+{\varvec{b}}{\varvec{x}}-{\varvec{c}}=0\) where a \(\ne 1\)

Students’ ability to solve equations of the form \({\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0\) was tested in Questions 4 and 5 on the written test. Both questions involved quadratics that were difficult to factorise as \(a\ne 1\) and the solutions were not whole numbers. It was a point of interest to observe whether factorisation was an efficient method of solving these types of quadratics. It was found that, where it was not an effective method, students tended to seek a solution by creatively violating algebraic conventions. An example of this is outlined in Fig. 9. All students attempted to solve Questions 4 and 5 via factorisation, however, only 9 out of 25, and 7 out of 25 students were able to factorise Question 4 and 5 respectively. When the quadratic was presented in a non-standard format, such as \(3{x}^{2}=-4x-1,\) predominantly students did not recognise the equation was a quadratic (i.e., tried to solve as if it were linear).

Fig. 9
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Example of student attempting to rearrange the quadratic to solve demonstrating algebraic misconceptions

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Students in their attempt to rearrange the quadratic equations demonstrated major algebraic misconceptions suggesting that both procedural fluency and conceptual understanding of algebraic processes was not deeply developed or connected to schema.

Summary of data

The detailed data above relates to specific errors. Table 4 provides an overview of the types of errors that students made in each question and the frequency of the error occurrence.

Table 4 Summary of error forms for each question and frequency of errors (n = 25)
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Overall, students experienced little success when attempting to solve quadratic equations via factorisation and fundamental algebra convention errors were evident.

Discussion

This study aimed to describe student performance in solving quadratic equations and identify patterns in the types of errors or misconceptions that were hindering students’ success. In this way, the findings from the study provide insight into the impact of the Australian mathematics curriculum’s design on student achievement in this school.

The study adds to the body of research on student’s performance on this topic area (e.g., Chaysuwan, 1996; Fonger et al., 2020; Huang & Kulm, 2012; Vaiyavutjamai & Clements, 2006; Vaiyavutjamai et al., 2005). Perhaps the most concerning finding in the error analysis was the prevalence of errors associated with violations of basic algebraic conventions. At this stage of the students’ study (senior mathematics), such errors are not expected particularly as the students in the sample had elected to study the more abstract mathematics of senior high school. If students had understood and been fluent in the critical algebra at some prior time, they had forgotten this by the time the study had been undertaken.

Regarding competency with quadratics, from the detailed analysis of student scripts, the research team identified five error patterns. These were then reclassified under four types of errors. This categorisation is outlined in Fig. 10.

Fig. 10
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Categorisation of errors

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Overall, it was apparent that students demonstrated either deficits in procedural aspects that led to incorrect solutions, or conceptual deficits that led them to select inappropriate procedures that were often incorrectly applied as they struggled to find a solution to the question. The analysis of student error patterns on scripts was supported with diagnostic interviews.

Interview data confirmed that students had deficits regarding the null factor law, as they were unable to indicate an understanding of this generalised rule whether or not they successfully applied the rule. Statements from students such as “I only know to pick the two numbers” indicated a procedural emphasis in prior teaching on how to solve a factorised quadratic, or that what may have been learnt had been forgotten. Overall, the fact that the null factor law was incorrectly applied, or simply not applied at all is an important finding in relation to teaching quadratic equations. Lacking understanding of the null factor law potentially explains why many students demonstrated subsequent misunderstandings regarding the nature of quadratics as they attempted to rearrange the equations to solve as if they were linear. Findings such as these support the need for mathematics concepts to be taught conceptually in the first instance, ensuring that new knowledge forms are linked to existing schema and have a logical basis.

The violation of basic algebraic procedures, while manifesting as procedural errors, is illustrative of a shallow conceptual understanding of the topic area. In this regard, the findings lend empirical evidence to the statement by Hiebert and Lefevre (1986) that students are “not fully competent in mathematics if either kind of knowledge is deficient or if they both have been acquired as separate entities” (p. 9). Knowledge of the null factor law and algebraic conventions has a conceptual basis, but senior secondary mathematics students are assumed to have obtained mastery of this law for fluent application. In other words, such prerequisite knowledge should have been procedural, and students at this level of study should have been fluent in the application of such a procedure. For most students, the curriculum as enacted by this school was not working. While there are potentially multiple reasons for this, a potential contributing factor is likely to be related to the mathematics curriculum structure.

Cognitive load theorists (e.g., Sweller, 2016; Van Merrienboer & Pass, 1990) provide a rationale for emphasising conceptual understanding in the first instance. First, if the subject matter is taught conceptually, it can be integrated into existing schema and thus be more easily remembered. The data suggests that knowledge of key mathematics concepts had not been resiliently linked in schematic form. As noted above, once conceptual knowledge is attained, the students must convert this to procedural knowledge (termed fluency in the Australian Curriculum). In this sense, procedure is not the same as blindly following a set of steps towards a solution, but rather that the steps are well embedded in long-term memory and can be readily applied with limited taxing of working memory. Hattie and Zierer (2019) recommend deliberate practice to develop fluency and, like cognitive load theorists, suggested that conscious, varied, spaced and regular practice fostered long-term memory retention. The ready recall of key facts and procedures is thought to reduce cognitive load (Van Merrienboer & Pass, 1990) and thereby enhance problem-solving. Rather than considering learning mathematics as a simple hierarchical progression, it is a progression through cycles of the form illustrated in Fig. 11.

Fig. 11
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The cyclic development of schema

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The approach depicted above is supported by what is understood about the development of conceptual and procedural understanding (Rosenshine, 2012; Sfard & Linchevski, 1994) and demonstrates that conceptualisation reaches a new depth at each interaction with the topic. The careful reorganisation of mental processes based on prior learning was central to the learning theories articulated by Piaget (1970) and Skemp (1976). Cognitive load theorists (e.g., Sweller, 2016; Van Merrienboer & Pass, 1990) provide further detail as to why it is important for students to conceptualise. When students understand and connect new knowledge to prior understandings, schema development is enriched and the development of schema counters forgetting (Sweller, 2016; Van Merrienboer & Pass, 1990). The data presents evidence that forgetting, or lack of conceptualisation occurred for students in this sample. Thus, the findings of this study indicate that the current structure of the curriculum enacted in the sample school was ineffective in developing students understanding or fluency with quadratic equations, or even much of the basic algebra from earlier years.

Recommendations and conclusion

This paper details students’ challenges with understanding and mastering concepts associated with quadratic equations. The recommendations arising from the findings from this study include reconsideration of whether the current Australian Curriculum is sufficiently designed to support a mastery approach. The curriculum enacted in the study school and the current Australian Curriculum is hierarchical and spiral in nature, as is common for mathematics curricula (ACARA, 2018). The analysis of the school and state curriculum recommendations suggested that a major limitation of the earlier study of quadratics was the extremely limited timeframe that was allocated to the topic area, as well as the fact that key concepts are revisited a year or more apart in the current curriculum design. The critical importance of time to conceptualise or build new knowledge (at the rates appropriate to individual) into schema is an assumption acknowledged by educational theories (e.g., Bruner, 1960; Piaget, 1970; Skemp, 1976).

The observation that many students exhibited limited proficiency with foundational algebra points to failures at potentially two key levels in the teaching and learning cycle modelled in Fig. 11. First, the manifestation of key conceptual errors related to the nature of quadratic equations suggests that initial conceptualisation of this mathematics topic was limited for significant numbers of students, and schema construction was limited. Unfortunately, the data on student errors does not tell us if this is a result of pedagogy that was not focused on conceptualisation, or simply a lack of time to enact appropriate pedagogy. The second significant finding was that students’ lacked understanding and fluency with basic algebraic conventions which ought to have been mastered in prior years. The two key recommendations are the integration of remediation practices, and greater time allocated for remediation and revision to aid remembering. The lack of time can be related to the nature of classroom discourse, however a lack of time due to curriculum structure is likely to be a contributing factor. The data from this study supports the reported concerns about the lack of time due to a crowded curriculum (Donnelly & Wiltshire, 2014; Snider, 2004). The findings indicate flaws in a curriculum design which touches on mathematics topics once a year, where mastery is also assumed in subsequent years.

The curriculum delivered in the school in this study differs from the most successful mathematics nations (e.g., China, Hong Kong, Singapore), which have structures in place to ensure that procedural and conceptual knowledge is integrated and well developed before embarking on conceptually new mathematics study (HKSARG, 2007; Singapore Ministry of Education, 2012). These structures include, firstly, the explicit direction to teach conceptually in curriculum documents and, secondly, more time dedicated to learning these concepts, and a greater emphasis on the importance of students developing fluency. The research into East Asian nations suggest that not only does the curriculum suggest greater time on task, but this is further increased with consistent homework engagement and tutor school attendance, features not always present in Australian middle schools (Norton, 2014; Norton & Zhang, 2013). With increased effective time on task and systematic revision, conceptualisation can be developed, and conceptual knowledge can build meaningfully on existing schema and stored in long-term memory (Sweller, 2016; Van Merrienboer & Pass, 1990). In short, in the study school, forgetting was not accounted for at two levels; initial conceptualisation and systematic revision.

The data and curriculum analysis in this study suggests that, in comparison with international curricula, it is apparent that a great deal of learning is expected of Australian students in a very condensed timeframe and that, for many students, this expectation was not fulfilled. This is at odds with what is known of brain functioning as expressed by cognitive load theorists and at odds with the design of effective curriculum that facilitates mastery.